YES Problem: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Proof: DP Processor: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) TDG Processor: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) graph: a__after#(0(),XS) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__after#(0(),XS) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(0(),XS) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(0(),XS) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(0(),XS) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) -> a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(s(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(s(N),cons(X,XS)) -> mark#(XS) -> mark#(from(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(s(X)) -> mark#(X) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(cons(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> mark#(X1) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(after(X1,X2)) -> mark#(X2) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> a__from#(mark(X)) a__after#(s(N),cons(X,XS)) -> mark#(N) -> mark#(from(X)) -> mark#(X) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> mark#(N) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(s(N),cons(X,XS)) -> mark#(XS) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) -> a__after#(0(),XS) -> mark#(XS) mark#(after(X1,X2)) -> mark#(X2) -> mark#(s(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X2) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X2) -> mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(after(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X1) mark#(after(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(s(X)) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X1) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(after(X1,X2)) -> mark#(X2) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(from(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) mark#(from(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(from(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) -> a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(s(X)) -> mark#(X) mark#(s(X)) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(s(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) mark#(s(X)) -> mark#(X) -> mark#(from(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(s(X)) -> mark#(X) a__from#(X) -> mark#(X) -> mark#(cons(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X1) a__from#(X) -> mark#(X) -> mark#(after(X1,X2)) -> mark#(X2) a__from#(X) -> mark#(X) -> mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) -> mark#(from(X)) -> mark#(X) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [a__after#](x0, x1) = x0 + x1, [mark#](x0) = x0, [a__from#](x0) = x0 + 0, [after](x0, x1) = 2x0 + x1 + 0, [a__after](x0, x1) = 2x0 + x1 + 0, [0] = 0, [cons](x0, x1) = x0 + x1 + 2, [from](x0) = x0 + 4, [s](x0) = x0 + 4, [mark](x0) = x0, [a__from](x0) = x0 + 4 orientation: a__from#(X) = X + 0 >= X = mark#(X) a__after#(0(),XS) = XS + 0 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= N = mark#(N) a__after#(s(N),cons(X,XS)) = N + X + XS + 4 >= N + XS = a__after#(mark(N),mark(XS)) mark#(from(X)) = X + 4 >= X = mark#(X) mark#(from(X)) = X + 4 >= X + 0 = a__from#(mark(X)) mark#(after(X1,X2)) = 2X1 + X2 + 0 >= X2 = mark#(X2) mark#(after(X1,X2)) = 2X1 + X2 + 0 >= X1 = mark#(X1) mark#(after(X1,X2)) = 2X1 + X2 + 0 >= X1 + X2 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 2 >= X1 = mark#(X1) mark#(s(X)) = X + 4 >= X = mark#(X) a__from(X) = X + 4 >= X + 4 = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 2 >= XS = mark(XS) a__after(s(N),cons(X,XS)) = 2N + X + XS + 6 >= 2N + XS + 0 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 4 >= X + 4 = a__from(mark(X)) mark(after(X1,X2)) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 2 >= X1 + X2 + 2 = cons(mark(X1),X2) mark(s(X)) = X + 4 >= X + 4 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = X + 4 >= X + 4 = from(X) a__after(X1,X2) = 2X1 + X2 + 0 >= 2X1 + X2 + 0 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Restore Modifier: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(N) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [a__after#](x0, x1) = 7x0 + x1 + 0, [mark#](x0) = x0, [a__from#](x0) = x0 + 7, [after](x0, x1) = 7x0 + x1 + 7, [a__after](x0, x1) = 7x0 + x1 + 7, [0] = 0, [cons](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 7, [s](x0) = x0 + 0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 7 orientation: a__from#(X) = X + 7 >= X = mark#(X) a__after#(0(),XS) = XS + 7 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = 7N + X + XS + 7 >= XS = mark#(XS) a__after#(s(N),cons(X,XS)) = 7N + X + XS + 7 >= N = mark#(N) a__after#(s(N),cons(X,XS)) = 7N + X + XS + 7 >= 7N + XS + 7 = a__after#(mark(N),mark(XS)) mark#(from(X)) = X + 7 >= X = mark#(X) mark#(from(X)) = X + 7 >= X + 7 = a__from#(mark(X)) mark#(after(X1,X2)) = 7X1 + X2 + 7 >= X2 = mark#(X2) mark#(after(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(s(X)) = X + 0 >= X = mark#(X) a__from(X) = X + 7 >= X + 7 = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 7 >= XS + 0 = mark(XS) a__after(s(N),cons(X,XS)) = 7N + X + XS + 7 >= 7N + XS + 7 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 7 >= X + 7 = a__from(mark(X)) mark(after(X1,X2)) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = X + 7 >= X + 7 = from(X) a__after(X1,X2) = 7X1 + X2 + 7 >= 7X1 + X2 + 7 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Restore Modifier: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [a__after#](x0, x1) = -3x0 + -3x1 + 0, [mark#](x0) = -5x0 + 0, [a__from#](x0) = -3x0 + 0, [after](x0, x1) = 2x0 + 2x1 + 5, [a__after](x0, x1) = 2x0 + 2x1 + 5, [0] = 3, [cons](x0, x1) = x0 + x1 + 5, [from](x0) = 2x0 + 5, [s](x0) = x0, [mark](x0) = x0 + 3, [a__from](x0) = 2x0 + 5 orientation: a__from#(X) = -3X + 0 >= -5X + 0 = mark#(X) a__after#(0(),XS) = -3XS + 0 >= -5XS + 0 = mark#(XS) a__after#(s(N),cons(X,XS)) = -3N + -3X + -3XS + 2 >= -5XS + 0 = mark#(XS) a__after#(s(N),cons(X,XS)) = -3N + -3X + -3XS + 2 >= -3N + -3XS + 0 = a__after#(mark(N),mark(XS)) mark#(from(X)) = -3X + 0 >= -5X + 0 = mark#(X) mark#(from(X)) = -3X + 0 >= -3X + 0 = a__from#(mark(X)) mark#(after(X1,X2)) = -3X1 + -3X2 + 0 >= -5X2 + 0 = mark#(X2) mark#(after(X1,X2)) = -3X1 + -3X2 + 0 >= -3X1 + -3X2 + 0 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = -5X1 + -5X2 + 0 >= -5X1 + 0 = mark#(X1) mark#(s(X)) = -5X + 0 >= -5X + 0 = mark#(X) a__from(X) = 2X + 5 >= 2X + 5 = cons(mark(X),from(s(X))) a__after(0(),XS) = 2XS + 5 >= XS + 3 = mark(XS) a__after(s(N),cons(X,XS)) = 2N + 2X + 2XS + 7 >= 2N + 2XS + 5 = a__after(mark(N),mark(XS)) mark(from(X)) = 2X + 5 >= 2X + 5 = a__from(mark(X)) mark(after(X1,X2)) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 5 >= X1 + X2 + 5 = cons(mark(X1),X2) mark(s(X)) = X + 3 >= X + 3 = s(mark(X)) mark(0()) = 3 >= 3 = 0() a__from(X) = 2X + 5 >= 2X + 5 = from(X) a__after(X1,X2) = 2X1 + 2X2 + 5 >= 2X1 + 2X2 + 5 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Restore Modifier: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(after(X1,X2)) -> mark#(X2) mark#(after(X1,X2)) -> a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [a__after#](x0, x1) = x0 + x1 + 0, [mark#](x0) = x0 + 0, [a__from#](x0) = x0 + 0, [after](x0, x1) = 4x0 + 4x1 + 5, [a__after](x0, x1) = 4x0 + 4x1 + 5, [0] = 0, [cons](x0, x1) = x0 + x1, [from](x0) = x0, [s](x0) = x0, [mark](x0) = x0 + 0, [a__from](x0) = x0 + 0 orientation: a__from#(X) = X + 0 >= X + 0 = mark#(X) a__after#(0(),XS) = XS + 0 >= XS + 0 = mark#(XS) a__after#(s(N),cons(X,XS)) = N + X + XS + 0 >= N + XS + 0 = a__after#(mark(N),mark(XS)) mark#(from(X)) = X + 0 >= X + 0 = mark#(X) mark#(from(X)) = X + 0 >= X + 0 = a__from#(mark(X)) mark#(after(X1,X2)) = 4X1 + 4X2 + 5 >= X2 + 0 = mark#(X2) mark#(after(X1,X2)) = 4X1 + 4X2 + 5 >= X1 + X2 + 0 = a__after#(mark(X1),mark(X2)) mark#(cons(X1,X2)) = X1 + X2 + 0 >= X1 + 0 = mark#(X1) mark#(s(X)) = X + 0 >= X + 0 = mark#(X) a__from(X) = X + 0 >= X + 0 = cons(mark(X),from(s(X))) a__after(0(),XS) = 4XS + 5 >= XS + 0 = mark(XS) a__after(s(N),cons(X,XS)) = 4N + 4X + 4XS + 5 >= 4N + 4XS + 5 = a__after(mark(N),mark(XS)) mark(from(X)) = X + 0 >= X + 0 = a__from(mark(X)) mark(after(X1,X2)) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = cons(mark(X1),X2) mark(s(X)) = X + 0 >= X + 0 = s(mark(X)) mark(0()) = 0 >= 0 = 0() a__from(X) = X + 0 >= X = from(X) a__after(X1,X2) = 4X1 + 4X2 + 5 >= 4X1 + 4X2 + 5 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Restore Modifier: DPs: a__from#(X) -> mark#(X) a__after#(0(),XS) -> mark#(XS) a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) SCC Processor: #sccs: 2 #rules: 6 #arcs: 72/49 DPs: a__after#(s(N),cons(X,XS)) -> a__after#(mark(N),mark(XS)) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [a__after#](x0, x1) = 5x0, [after](x0, x1) = 4x1 + 0, [a__after](x0, x1) = 4x1 + 0, [0] = 3, [cons](x0, x1) = x1 + 3, [from](x0) = 3, [s](x0) = 1x0, [mark](x0) = x0, [a__from](x0) = 3 orientation: a__after#(s(N),cons(X,XS)) = 6N >= 5N = a__after#(mark(N),mark(XS)) a__from(X) = 3 >= 3 = cons(mark(X),from(s(X))) a__after(0(),XS) = 4XS + 0 >= XS = mark(XS) a__after(s(N),cons(X,XS)) = 4XS + 7 >= 4XS + 0 = a__after(mark(N),mark(XS)) mark(from(X)) = 3 >= 3 = a__from(mark(X)) mark(after(X1,X2)) = 4X2 + 0 >= 4X2 + 0 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = X2 + 3 >= X2 + 3 = cons(mark(X1),X2) mark(s(X)) = 1X >= 1X = s(mark(X)) mark(0()) = 3 >= 3 = 0() a__from(X) = 3 >= 3 = from(X) a__after(X1,X2) = 4X2 + 0 >= 4X2 + 0 = after(X1,X2) problem: DPs: TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Qed DPs: mark#(from(X)) -> mark#(X) mark#(from(X)) -> a__from#(mark(X)) a__from#(X) -> mark#(X) mark#(cons(X1,X2)) -> mark#(X1) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Arctic Interpretation Processor: dimension: 1 usable rules: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) interpretation: [mark#](x0) = x0, [a__from#](x0) = x0, [after](x0, x1) = x0 + x1 + -8, [a__after](x0, x1) = x0 + x1 + -8, [0] = 2, [cons](x0, x1) = 1x0 + x1, [from](x0) = 1x0, [s](x0) = x0, [mark](x0) = x0, [a__from](x0) = 1x0 orientation: mark#(from(X)) = 1X >= X = mark#(X) mark#(from(X)) = 1X >= X = a__from#(mark(X)) a__from#(X) = X >= X = mark#(X) mark#(cons(X1,X2)) = 1X1 + X2 >= X1 = mark#(X1) mark#(s(X)) = X >= X = mark#(X) a__from(X) = 1X >= 1X = cons(mark(X),from(s(X))) a__after(0(),XS) = XS + 2 >= XS = mark(XS) a__after(s(N),cons(X,XS)) = N + 1X + XS + -8 >= N + XS + -8 = a__after(mark(N),mark(XS)) mark(from(X)) = 1X >= 1X = a__from(mark(X)) mark(after(X1,X2)) = X1 + X2 + -8 >= X1 + X2 + -8 = a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) = 1X1 + X2 >= 1X1 + X2 = cons(mark(X1),X2) mark(s(X)) = X >= X = s(mark(X)) mark(0()) = 2 >= 2 = 0() a__from(X) = 1X >= 1X = from(X) a__after(X1,X2) = X1 + X2 + -8 >= X1 + X2 + -8 = after(X1,X2) problem: DPs: a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Restore Modifier: DPs: a__from#(X) -> mark#(X) mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) SCC Processor: #sccs: 1 #rules: 1 #arcs: 17/4 DPs: mark#(s(X)) -> mark#(X) TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) Size-Change Termination Processor: DPs: TRS: a__from(X) -> cons(mark(X),from(s(X))) a__after(0(),XS) -> mark(XS) a__after(s(N),cons(X,XS)) -> a__after(mark(N),mark(XS)) mark(from(X)) -> a__from(mark(X)) mark(after(X1,X2)) -> a__after(mark(X1),mark(X2)) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(s(X)) -> s(mark(X)) mark(0()) -> 0() a__from(X) -> from(X) a__after(X1,X2) -> after(X1,X2) The DP: mark#(s(X)) -> mark#(X) has the edges: 0 > 0 Qed